Geometric characterizations of the representation type of hereditary algebras and of canonical algebras. (English) Zbl 1252.16014
Let \(k\) be an algebraically closed field of characteristic \(0\). The guiding problem of the paper is to find characterizations of the representation type in terms of properties of the associated geometric objects. In the paper in question, the geometric objects studied are rational invariants.
More precisely, let \(\Lambda\) be a finite dimensional \(k\)-algebra and \(\mathbf d\) a dimension vector. One defines a variety \(\text{mod}_\Lambda (\mathbf d)\), called the variety of \(\Lambda\)-modules with dimension vector \(\mathbf d\), in such a way that it possesses an action of an algebraic group \(\text{GL}(\mathbf d)\), which is natural in the sense that the orbits correspond to the isomorphism classes of the \(\Lambda\)-modules with dimension vector \(\mathbf d\). Given an irreducible component \(C\) one may study the field \(k(C)^{\mathbf d}\) of the rational invariants with respect to this action. An irreducible component is called indecomposable if it contains a dense subset of indecomposable modules.
The main result of the paper states that if \(\Lambda\) is either hereditary or canonical (in the sense of Ringel), then \(\Lambda\) is tame if and only if for each indecomposable irreducible component \(C\) the field \(k(C)^{\mathbf d}\) is either \(k\) or \(k (t)\). In the case of hereditary algebras the author gives also two alternative characterizations of the tame representation type.
Despite proving the above mentioned interesting results the author introduces a useful method allowing to compare the fields of rational invariants for different algebras. In particular, he uses this method in order to reduce the proof in the case of the canonical algebras to the Kronecker quiver.
More precisely, let \(\Lambda\) be a finite dimensional \(k\)-algebra and \(\mathbf d\) a dimension vector. One defines a variety \(\text{mod}_\Lambda (\mathbf d)\), called the variety of \(\Lambda\)-modules with dimension vector \(\mathbf d\), in such a way that it possesses an action of an algebraic group \(\text{GL}(\mathbf d)\), which is natural in the sense that the orbits correspond to the isomorphism classes of the \(\Lambda\)-modules with dimension vector \(\mathbf d\). Given an irreducible component \(C\) one may study the field \(k(C)^{\mathbf d}\) of the rational invariants with respect to this action. An irreducible component is called indecomposable if it contains a dense subset of indecomposable modules.
The main result of the paper states that if \(\Lambda\) is either hereditary or canonical (in the sense of Ringel), then \(\Lambda\) is tame if and only if for each indecomposable irreducible component \(C\) the field \(k(C)^{\mathbf d}\) is either \(k\) or \(k (t)\). In the case of hereditary algebras the author gives also two alternative characterizations of the tame representation type.
Despite proving the above mentioned interesting results the author introduces a useful method allowing to compare the fields of rational invariants for different algebras. In particular, he uses this method in order to reduce the proof in the case of the canonical algebras to the Kronecker quiver.
Reviewer: Grzegorz Bobiński (Toruń)
MSC:
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16G20 | Representations of quivers and partially ordered sets |
| 16G10 | Representations of associative Artinian rings |
| 14R20 | Group actions on affine varieties |
| 14L24 | Geometric invariant theory |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
Keywords:
quivers; canonical algebras; hereditary algebras; rational invariants; representation types; varieties of modules; tame representation typeReferences:
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